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Voorbeeld presentatie BeamerTU Huisstijl
Voornaam Achternaam, Afdeling30 maart 2010
() Voorbeeld presentatie Beamer 1 / 16
Outline
1 First sectionSection 1 - subsection 1Section 1 - subsection 2Section 1 - subsection 3
2 Second sectionSection 2 - subsection 1Section 2 - last subsection
() Voorbeeld presentatie Beamer 2 / 16
Next subsection
1 First sectionSection 1 - subsection 1Section 1 - subsection 2Section 1 - subsection 3
2 Second sectionSection 2 - subsection 1Section 2 - last subsection
() Voorbeeld presentatie Beamer 3 / 16
Section 1 - subsection 1 - page 1
Example
0 1 2 3 4 5 6 7 8n
0123456u(n)
u(n) = [3, 1, 4]n
() Voorbeeld presentatie Beamer 4 / 16
Section 1 - subsection 1 - page 2
Definition
Let n be a discrete variable, i.e. n ∈ Z. A 1-dimensionalperiodic number is a function that depends periodically on n.
u(n) = [u0, u1, . . . , ud−1]n =
u0 if n ≡ 0 (mod d)
u1 if n ≡ 1 (mod d)...
ud−1 if n ≡ d − 1 (mod d)
d is called the period.
() Voorbeeld presentatie Beamer 5 / 16
Section 1 - subsection 1 - page 3
Example
f (n) = −[12 , 1
3
]nn2 + 3n − [1, 2]n
=
{−1
3n2 + 3n − 2 if n ≡ 0 (mod 2)
−12n
2 + 3n − 1 if n ≡ 1 (mod 2)
0 1 2 3 4 5 6 7n
−3
−2
−1
0
1
2
3
4
5f(n)=
−[ 1 2,1 3
] nn2+3n
−[1,2] n
−12n2 + 3n− 1
−13n2 + 3n− 2
() Voorbeeld presentatie Beamer 6 / 16
Section 1 - subsection 1 - page 4
Definition
A polynomial in a variable x is a linear combination of powersof x :
f (x) =
g∑i=0
cixi
Definition
A quasi-polynomial in a variable x is a polynomial expressionwith periodic numbers as coefficients:
f (n) =
g∑i=0
ui (n)ni
with ui (n) periodic numbers.
() Voorbeeld presentatie Beamer 7 / 16
Section 1 - subsection 1 - page 4
Definition
A polynomial in a variable x is a linear combination of powersof x :
f (x) =
g∑i=0
cixi
Definition
A quasi-polynomial in a variable x is a polynomial expressionwith periodic numbers as coefficients:
f (n) =
g∑i=0
ui (n)ni
with ui (n) periodic numbers.
() Voorbeeld presentatie Beamer 7 / 16
Next subsection
1 First sectionSection 1 - subsection 1Section 1 - subsection 2Section 1 - subsection 3
2 Second sectionSection 2 - subsection 1Section 2 - last subsection
() Voorbeeld presentatie Beamer 8 / 16
Section 1 - subsection 2 - page 1
Example
0 1 2 3 4 5 6 7x
0
1
2
3
4
5
6
7
y
p =3
x + y 6 p
p f (p)
3 54 85 106 13
5
2p +
[−2,
−5
2
]
p
() Voorbeeld presentatie Beamer 9 / 16
Section 1 - subsection 2 - page 1
Example
0 1 2 3 4 5 6 7x
0
1
2
3
4
5
6
7
y
p =4
x + y 6 p
p f (p)
3 54 85 106 13
5
2p +
[−2,
−5
2
]
p
() Voorbeeld presentatie Beamer 9 / 16
Section 1 - subsection 2 - page 1
Example
0 1 2 3 4 5 6 7x
0
1
2
3
4
5
6
7
y
p =5
x + y 6 p
p f (p)
3 54 85 106 13
5
2p +
[−2,
−5
2
]
p
() Voorbeeld presentatie Beamer 9 / 16
Section 1 - subsection 2 - page 1
Example
0 1 2 3 4 5 6 7x
0
1
2
3
4
5
6
7
y
p =6
x + y 6 p
p f (p)
3 54 85 106 13
5
2p +
[−2,
−5
2
]
p
() Voorbeeld presentatie Beamer 9 / 16
Section 1 - subsection 2 - page 1
Example
0 1 2 3 4 5 6 7x
0
1
2
3
4
5
6
7
y
p =6
x + y 6 p
p f (p)
3 54 85 106 13
5
2p +
[−2,
−5
2
]
p
() Voorbeeld presentatie Beamer 9 / 16
Section 1 - subsection 2 - page 2
The number of integer points in a parametric polytope Pp
of dimension n is expressed as a piecewise aquasi-polynomial of degree n in p (Clauss and Loechner).
More general polyhedral counting problems:Systems of linear inequalities combined with ∨,∧,¬, ∀, or∃ (Presburger formulas).
Many problems in static program analysis can be expressedas polyhedral counting problems.
() Voorbeeld presentatie Beamer 10 / 16
Section 1 - subsection 2 - page 2
The number of integer points in a parametric polytope Pp
of dimension n is expressed as a piecewise aquasi-polynomial of degree n in p (Clauss and Loechner).
More general polyhedral counting problems:Systems of linear inequalities combined with ∨,∧,¬, ∀, or∃ (Presburger formulas).
Many problems in static program analysis can be expressedas polyhedral counting problems.
() Voorbeeld presentatie Beamer 10 / 16
Section 1 - subsection 2 - page 2
The number of integer points in a parametric polytope Pp
of dimension n is expressed as a piecewise aquasi-polynomial of degree n in p (Clauss and Loechner).
More general polyhedral counting problems:Systems of linear inequalities combined with ∨,∧,¬, ∀, or∃ (Presburger formulas).
Many problems in static program analysis can be expressedas polyhedral counting problems.
() Voorbeeld presentatie Beamer 10 / 16
Next subsection
1 First sectionSection 1 - subsection 1Section 1 - subsection 2Section 1 - subsection 3
2 Second sectionSection 2 - subsection 1Section 2 - last subsection
() Voorbeeld presentatie Beamer 11 / 16
Section 1 - subsection 3 - page 1
A picture made with the package TiKz
Example
1
() Voorbeeld presentatie Beamer 12 / 16
Next subsection
1 First sectionSection 1 - subsection 1Section 1 - subsection 2Section 1 - subsection 3
2 Second sectionSection 2 - subsection 1Section 2 - last subsection
() Voorbeeld presentatie Beamer 13 / 16
Section 2 - subsection 1 - page 1
Problem
This page gives an example with numbered bullets (enumerate)in an ”Example”window:
Example
Discrete domain ⇒ evaluate in each pointNot possible for
1 parametric domains
2 large domains (NP-complete)
() Voorbeeld presentatie Beamer 14 / 16
Section 2 - subsection 1 - page 1
Problem
This page gives an example with numbered bullets (enumerate)in an ”Example”window:
Example
Discrete domain ⇒ evaluate in each pointNot possible for
1 parametric domains
2 large domains (NP-complete)
() Voorbeeld presentatie Beamer 14 / 16
Next subsection
1 First sectionSection 1 - subsection 1Section 1 - subsection 2Section 1 - subsection 3
2 Second sectionSection 2 - subsection 1Section 2 - last subsection
() Voorbeeld presentatie Beamer 15 / 16
Last page
Summery
End of the beamer demowith a TUDelft lay-out.
Thank you!
() Voorbeeld presentatie Beamer 16 / 16